Basic calculus

Analytic Geometry 2
 Circle
• A circle is a locus of a moving point P(x,y) which moves
so that its distance called radius from a fixed point called
center is always constant.
 Standard Equation of a Circle with Center at (h, k)

• Where (h, k) = location of center of the circle

r = radius of the circle
• Example 1: What is the standard equation of a circle with
center at (4, -2) and radius of length of 3?
Ans.
Try Solving This!
• What is the standard equation of a circle with center at (2,
-6) and the diameter of 10?
• What is the equation of a circle that has its center at (2, -
3) and passes through point (-2, 0)?
• Find the equation of a circle tangent to the line
 General Equation of a Circle with Center at (h, k)

• Note: The coefficient of x2 and y2 must be equal to 1. If

the coefficients are not yet equal to 1, simplify it first.
• Coordinates of the center:

• Radius of the circle:

• Area of the circle:
• Circumference of the circle:
• Example 2: A circle has the equation of
Quiz:

• Determine the coordinates of the center of the circle,

Parabola
• A parabola is a locus of a moving point P(x,y) which
moves so that its distance from a fixed point called focus
is always equal to its distance from a fixed line called
directrix.

• Note: d1 = d2
• V = vertex; F = focus
• a = distance from focus to vertex, also equal to the distance of directrix to
vertex.
• = commonly called focal length
• Latus Rectum - a chord drawn through the focus,
perpendicular to the axis of symmetry of the curve.
• Length of Latus Rectum (LR) on a parabola:

• Eccentricity (e) - the ratio of the distance of the moving

point from the focus (fixed point) to its distance from the
directrix (fixed line).
• Note: For parabola, since those distances mentioned are
equal, thus its eccentricity, e = 1.
• Main axis or axis of symmetry - the line that passes the
focus and vertex on a parabola.
Standard Equation of a Parabola with Vertex at (h, k)

opening to the right

opening to the left
opening up
opening down
• Example 4: Find the equation of a parabola with vertex at
(-2, 3) and focus at (-4, 3).
Ans.
• Example 5: Find the equation of a parabola with focus at
(4, 2) and directrix of y = -4.
Ans.
• Example 6: A parabola has its focus at (4, 3). If the
directrix of this parabola is the line y = 5, solve the length
of its latus rectum.
Ans. 16 unit
Seatwork
• Find the equation of a parabola with vertex at (3, -2) and
the ends of the latus rectum at (-2, 1/2) and (8, 1/2).
• Find the equation of a parabola with focus at (2, -3) and a
directrix at x = 4.
• A parabola has its vertex at the origin, its main axis is
along the x-axis and it passes through the point (-3, 6).
Solve its focal length.
• A cable suspended from supports which are 200 ft apart
has a sag of 50 ft. If cable hangs in the form of a parabola,
find its equation,taking the origin at the lowest point.
 General Equation of a Parabola with Vertex at (h, k)
• Parabola opens to the left or to the right

• Note: If D is positive, the parabola opens to the left.

If D is negative, the parabola opens to the right.
• Coordinates of the vertex:
• Length of latus rectum:
• Focal length:
• Parabola opens upward or downward

• Note: If E is positive, the parabola opens dwon.

If E is negative, the parabola opens up.
• Coordinates of the vertex:
• Length of latus rectum:
• Focal length:
• Example 7: A parabola has the equation
Seatwork
• Solve the coordinates of the focus of a parabola whose
equation is
Area Enclosed by a parabolic Segment

where: b = base
h = height
• Example 8: Solve the area bounded by the parabola
whose equation is and its latus rectum.
Ans. 24 sq. units

Relation Between Bases and Heights

on a Parabola
• Example 9: A satellite disk in storage, has a parabolic
cross section and is resting on its vertex. A point on its rim
is 4 ft high and is 6 ft horizontally from the vertex. How
high is a point which is 3 ft horizontally from the vertex?
Ans. 1 ft
• Example 10: A cable suspended from supports that are
the same height and 600 ft apart has a sag of 100 ft. If the
cable hangs in the forms of a parabola taking the origin at
the lowest point, find the width of the cable at a height of
50 ft above the lowest point.
Ans. 424.26 ft
Seatwork
• A parabola with a vertical axis has its vertex at the origin
and passes through point (7, 7). The parabola intersects a
line y = 6 at two points. The length of the segment joining
these points is equal to ______.

• The entrance to a formal garden is spanned by a

parabolic arch 10 ft high and 10 ft across at its base. How
wide should the walk through the center of the arch be if a
minimum clearance of 7 ft is wanted above the walk?
 Ellipse

• An ellipse is a locus of a moving point P(x,y) which moves

so that the sum of its distances from two fixed points
called the foci is constant and is equal to the length of its
major axis.
• Note: d1 + d2 = 2a
• a = length of semi-major axis
• b = length of semi-minor axis
• c = distance of focus to center =
• d = distance of directrix to center =
• Length of major axis: 2a
• Length of minor axis: 2b
• Distance between foci: 2c
• Eccentricity: (for ellipse, e˂1)
• Note: An ellipse has two latus rectum.
• Length of latus rectum:
• Latera recta - plural word for latus rectum. The length of
the latera recta means the length of the two latus rectum.
• Area of an ellipse:

• Perimeter of an ellipse:
Standard Equation of an Ellipse with Center at (h, k)

• Major axis is horizontal: Major axis is vertical:

• Example 11: Find the equation of an ellipse with center at
(5, 4), axis is horizontal, major axis is 16 and minor axis is
10.
Ans.
• Example 12: Find the equation of an ellipse with foci (2, 1)
and (2, -1) and semi-major axis = 2.
Ans.
• Example 13: Find the area of an ellipse for which the foci
are (-8, 2) and (4, 2) and the eccentricity is 2/3.
Ans. 189.7 sq. units
Quiz!
• Find the equation of an ellipse with center at (2, 0), focus
(5, 0) and semi-minor axis = 4.
• Find the equation of an ellipse if the foci are located at (-5,
-3) and (-5, -7) and the length of latus rectum is 6.
• Solve the eccentricity of an ellipse whose major axis is
twice as long as its minor axis.
• Solve the area of the ellipse with foci at (4, -2) and (10, -
2) and one vertex at (12, -2)
 General Equation of an Ellipse with Center at (h, k)

• Coordinates of center:
• Major axis is horizontal (A˂C)
• Length of semi-major axis:
• Length of semi-minor axis:
• Major axis is vertical (A˃C)
• Length of semi-major axis:
• Length of semi-minor axis:
• Example 14: An ellipse has the equation
 Assignment

• Solve the length of latus rectum of the ellipse whose

equation is
 Apogee and Perigee
• The path of a satellite revolving around the earth is an
elliptical path. The path of the earth revolving around the
sun is also an elliptical path. The path of the moon
revolving around the planet is also an elliptical path.
• Apogee is the farthest distance of a satellite from the
earth surface while perigee is the shortest distance.

r = radius of the planet

• Example 15: An earth satellite’s orbit is an ellipse with the
earth at the focus. The satellite has an apogee of 45,000
km and a perigee of 5,000 km. Assuming the radius of
earth equal to 6400 km, solve the eccentricity of the
elliptical orbit.
Ans. 0.637
Assignment!

• The earth’s orbit is an ellipse, with the sun at the focus.

The length of major axis is 180,000,000 miles and the
eccentricity is 0.02. Determine the perigee of the earth to
the sun.
 Hyperbola
• A hyperbola is a locus of a moving point P(x,y) which
moves so that the difference of its distances from two
fixed points called foci is constant and is equal to length
of its transverse axis.
• Asymptote - a line that draws increasingly nearer to a
curve without ever meeting it.
• a = length of semi-transverse axis
• b = length of semi-conjugate axis
• c = distance of focus to center =
• d = distance of directrix to center =
• Length of transverse axis: 2a
• Length of conjugate axis: 2b
• Distance between foci: 2c
• Eccentricity: (for hyperbola, e˃1)
• Length of latus rectum:

• Note: A hyperbola has two latus rectum

 Standard Equation of a Hyperbola
with Center at (h, k)
• Transverse axis is horizontal:

• Transverse axis is vertical:

• Example 16: Find the equation of a hyperbola with center
at (1, 1), vertex (3, 1) and eccentricity of 2.
Ans.

• Example 17: Find the equation of a hyperbola whose foci

at (0, 6) and (0, -6) and the length of the transverse axis
is 10.
Ans.
General Equation of a Hyperbola with Center at (h, k)
• Transverse axis is horizontal

• Length of semi-major axis:

• Length of semi-minor axis:
• Coordinates of the center:
• Transverse axis is vertical
• Example 18: A hyperbola has the equation
 Equation of the Asymptote of a Hyperbola
• Using point-slope form for the equation of a line, the
equation of the asymptote is,
±
Where: (h, k) → coordinates of the center
m = slope
• Example 19: Find the equation of the asymptotes of the
hyperbola,
±
 Conic Sections
• A conic section is produced by passing a cutting plane
through a right circular cone.

• When cutting plane is inclined but not passing the base → ellipse
• When cutting plane is parallel to the base → circle
• When cutting plane is parallel to the element or lateral surface of
the cone → parabola
• When cutting plane is at an angle closer to the axis of cone
(almost perpendicular) → hyperbola
• If B = 0, the principal axis is parallel to the standard x and
y axis.
• If either A or C is zero, the graph is a parabola
• If A & C have opposite signs, the graph is a hyperbola
• If A C and same signs, the graph is an ellipse
• If A = C and same signs, the graph is a circle
• If B 0, the principal axis is inclined from the standard x
and y axis.
• If
• Example 21: The equation
represents what type of conic section?
Ans. parabola.

Assignment!
• A curve whose equation is is
what type of conic section?

• A curve whose equation is

is what type of conic section?